Lecture on Using the Distance Formula and Circle Area Calculation

Distance Formula

• Purpose: Calculate the distance between two points on a plane.
• Formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
  • Variables:
    • (x_1, y_1): Coordinates of the first point.
    • (x_2, y_2): Coordinates of the second point.

Example 1

• Points: A(1, 2) and B(9, 17)
• Steps:
  • Identify coordinates: (x_1 = 1, y_1 = 2, x_2 = 9, y_2 = 17)
  • Calculate differences: (x_2 - x_1 = 8, y_2 - y_1 = 15)
  • Calculate squares: (8^2 = 64, 15^2 = 225)
  • Sum of squares: (64 + 225 = 289)
  • Distance: (\sqrt{289} = 17)
• Result: Distance = 17 units

Example 2

• Points: C(5, -16) and D(-2, 8)
• Steps:
  • Identify coordinates: (x_1 = 5, y_1 = -16, x_2 = -2, y_2 = 8)
  • Calculate differences: (-2 - 5 = -7, 8 - (-16) = 24)
  • Calculate squares: ((-7)^2 = 49, 24^2 = 576)
  • Sum of squares: (49 + 576 = 625)
  • Distance: (\sqrt{625} = 25)
• Result: Distance = 25 units

Calculating Area of a Circle

• Given: Center at (2, 1) and point P(6, 4) on circle
• Objective: Find the area of the circle

Steps

1. Find Radius using Distance Formula:

   • Use center and point on circle: Center (2, 1), Point P(6, 4)
   • Identify coordinates: (x_1 = 2, y_1 = 1, x_2 = 6, y_2 = 4)
   • Calculate differences: (6 - 2 = 4, 4 - 1 = 3)
   • Calculate squares: (4^2 = 16, 3^2 = 9)
   • Sum of squares: (16 + 9 = 25)
   • Radius: (\sqrt{25} = 5)

2. Calculate Area:

   • Formula: Area = (\pi r^2)
   • Radius (r = 5)
   • Area: (\pi \times 5^2 = 25\pi)
   • Result: Area = 25(\pi) square units

These are the key points and examples on using the distance formula and calculating the area of a circle based on given coordinates.